A couple thoughts to make your efforts a success from things I have noticed in your pictures, my experiences, and are in no particular order..
1. P
lug up all the open areas you have and clean that engine. put dip stick filler back in, put plug in, cover the carb area with plastic. I use Spray 9 or Tough Stuff and it will clean and degrease everything, sprays off with water. I like the concentrated Spray 9, get from AMZN, you can make cleaner spray bottles with this diluted.
2. Your air box looks crooked in the pic. Tells me your carburetor mounting bolts may be loose. Get yourself a new air filter, plug, and change the oil.
3. your carb is a "chokeless version that depends on the primer bulb to squirt gas into the carb for initial start. with the cover off the air box look inside and be sure when you press the primer bulb you see a squirt of gas in the throat of the carb. If not you might need a gasket or some ferreting to find out why. If airbox / carb bolts are found to be loose, then that would cause this.
4. Check every bolt on that engine for being tight. especially the carb, air box runner that goes to the head. Everything. Don't go crazy but just check em for being run down snug.
5. Be really careful when putting on the new coil. Those mounting ears are very easy to break off (then you are done) as you are tightening steel screws into aluminum. clean, inspect, test run in both screws when mounting the new coil. I would clean the threaded holes as well and maybe use a lil WD in the holes to make it a smooth install., take some sandpaper and sand the top contact area on each post as you need a good ground between your new coil and engine. Be sure to test fit any new mounting screws that may come with new coil as the threads could be different. Do not force anything!
6. It can be easy to mount a coil backwards. be sure to get the coil oriented properly and gap the coil with a business card between coils and flywheel magnets or to about 10 thousand if you want to get crazy. Just need a gap, no rub.
7. other poster is correct; you can try pretty much any Briggs coil as they are pretty much all the same. I have a few that I keep for testing and such.
OK, I'm done ..... Apologies for singing to the choir but hey it is Saturday and I just had to get it out.